Global forward self-similar solutions of the spatially inhomogeneous Boltzmann equation for cutoff Maxwell molecules
Abstract
For every fixed m>1, we construct nonzero global forward self-similar mild solutions of the three-dimensional spatially inhomogeneous Boltzmann equation for Maxwell molecules. We assume that the angular kernel is bounded and satisfies a weighted integrability condition at the grazing and head-on angles. The initial data are homogeneous under the kinetic scaling, singular at the origin in phase space, and have infinite mass. Their size is small in a weighted norm invariant under this scaling. We first prove a bilinear estimate for the gain operator in this norm, using free-transport coordinates and the orthogonality of the collision displacements. This estimate gives a global solution of the equation without the loss term, which bounds the lower and upper approximations in the Kaniel--Shinbrot iteration. We prove that their difference tendsto zero to arbitrarily high order at the initial time on a fixed annulus in phase space. The argument uses a finite number of iterations of the collision estimate and stops characteristics when they leave a spatial annulus. Self-similarity then gives finite mass and a finite first absolute velocity moment for the difference. Comparing its mass balance with its scaling law shows that the two limits agree. The resulting solution attains its initial trace locally in L∞ away from the origin in phase space and has infinite mass at every positive time.
Related
- Well-posedness for the spatially inhomogeneous Fermi-Dirac-Fokker-Planck equation
- Weakly inhomogeneous solutions to a nonlinear Vlasov--Fokker--Planck equation in a domain
- Global smooth solutions to the inhomogeneous Landau-Fermi-Dirac equation
- Runaway avalanches in plasmas with external electric fields: spatially inhomogeneous case in a perturbation framework
- Fock-Space Formulation of the Boltzmann Collision Operator for Maxwell Molecules
BibTeX
@misc{nguyen2026global,
title = {{Global forward self-similar solutions of the spatially inhomogeneous Boltzmann equation for cutoff Maxwell molecules}},
author = {Quoc-Hung Nguyen and Jiaqi Yang and Tong Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28285},
eprint = {2609.28285},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28285},
}
RIS
TY - UNPB
TI - Global forward self-similar solutions of the spatially inhomogeneous Boltzmann equation for cutoff Maxwell molecules
AU - Quoc-Hung Nguyen
AU - Jiaqi Yang
AU - Tong Yang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28285
UR - https://arxiv.org/abs/2609.28285
AB - For every fixed m>1, we construct nonzero global forward self-similar mild solutions of the three-dimensional spatially inhomogeneous Boltzmann equation for Maxwell molecules. We assume that the angular kernel is bounded and satisfies a weighted integrability condition at the grazing and head-on angles. The initial data are homogeneous under the kinetic scaling, singular at the origin in phase space, and have infinite mass. Their size is small in a weighted norm invariant under this scaling. We first prove a bilinear estimate for the gain operator in this norm, using free-transport coordinates and the orthogonality of the collision displacements. This estimate gives a global solution of the equation without the loss term, which bounds the lower and upper approximations in the Kaniel--Shinbrot iteration. We prove that their difference tendsto zero to arbitrarily high order at the initial time on a fixed annulus in phase space. The argument uses a finite number of iterations of the collision estimate and stops characteristics when they leave a spatial annulus. Self-similarity then gives finite mass and a finite first absolute velocity moment for the difference. Comparing its mass balance with its scaling law shows that the two limits agree. The resulting solution attains its initial trace locally in L^∞ away from the origin in phase space and has infinite mass at every positive time.
ER -